Thermocapillary manipulation of microfluidic droplets: Theory and applications

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Thermocapillary manipulation of microfluidic droplets:

Theory and applications

François Gallaire, Charles Baroud, Jean-Pierre Delville

To cite this version:

François Gallaire, Charles Baroud, Jean-Pierre Delville. Thermocapillary manipulation of

microflu-idic droplets: Theory and applications. International Journal of Heat and Technology, International

Information and Engineering Technology Association, 2008, 26 (1), pp.161-166. �hal-00384745�

THERMOCAPILLARY MANIPULATION OF MICROFLUIDIC

DROPLETS: THEORY AND APPLICATIONS

Fran¸cois Gallaire

, Charles N. Baroud

, Jean-Pierre Delville

1

Laboratoire JA Dieudonn´e, Universit´e de Nice Sophia-Antipolis

2

LadHyX, Ecole Polytechnique, 91128 Palaiseau Cedex

3

CPMOH, Universt´e de Bordeaux I, 351 Cours de la lib´eration, F-33405 Talence Cedex

September 12, 2007

Abstract

It was recently demonstrated by our group that a focused laser beam could be used to produce a net force on a moving microfluidic drop. The aim of the paper is to establish a scaling law for this net force by a examining the closely related but simpler situation of a very thin stationary circular drop of fixed shape submitted to a thermocapillary (Marangoni) stress. This leads us to recall the depth-averaged model for a microfluidic pancake-like undeformable drop submitted to a thermocapillary forcing. Our numerical method to solve the associated equations is then introduced and validated. In the case of a localized heating and for an ‘inverse’ Marangoni effect (i.e. the surface tension increases with temperature) mimicking the experimental situation of a focused laser beam impinging on a surfactant laden water-oil interface, the flow field is computed and compared to experimental observations. The viscous shear stresses (normal and tangential) and the pressure force are then computed on the interface, yielding a simple expression for the total force acting on the droplet. Further numerical investigations are conducted and enable us to propose a scaling law for the net force combining all pertinent parameters.

1. INTRODUCTION

Effective actuation of microfluidic drops may appear as simple, but when both inertial and buoyancy effects are negligible, the manipula-tion of water in oil drops turns out to be diffi-cult to achieve. Furthermore, several more com-plicated operations are necessary if one wishes to use droplets as microreactors in microfluidic lab-on-a-chip systems. These operations include the ability to create a reaction (fusion of drops and mixing of a drop’s content), test the results (sort drops), or sample their contents by divid-ing them asymmetrically.

We have recently demonstrated that a local-ized heating from a laser source on the surface of the drop may be used to produce these op-erations through the thermocapillary effect, by which the surface tension of an interface varies

with the temperature. In our recent

experi-ments [1], we showed that a water in oil drop, transported by the external oil flow, may be blocked if submitted to a focused laser spot. This blocking may last a few seconds, imply-ing that the force due to the drag from the oil is

balanced (at least temporarily) by a force which is generated by the laser heating. This block-ing force was then combined with the geome-try of the micro-channel to succeed in most of the aforementioned operations (merging, sorting and dividing).

The effect of temperature gradients on drops has been studied since the 1950’s because of its applications in microgravity situations [2, 3] and has been recently revived by the applications in microfluidics [4, 5]. The generally accepted de-scription [2, 4] is that surface tension decreases with temperature. Therefore if a drop is placed in a temperature gradient, it will experience flow along the interface from the hot to the cold side. In a low Reynolds number situation, this flow very rapidly (immediately for Re → 0) produces a matching flow in the outer fluid from the hot to the cold region. Therefore by conservation of matter, the drop must “swim” towards the hot side.

This description is not compatible with our observations as deduced from figure 1 which de-picts a cross-shaped microchannel in which oil

Laser off

Laser on

_Laser

(d)

(e)

(f)

(a)

(b)

(c)

Figure 1: The water interface is temporarily blocked when it crosses the path of the focused laser in response to a force directed away from the hot spot, resulting in the production of a larger drop.

flows from the top and bottom branches and wa-ter from the left branch. Wawa-ter drops in oil are produced and flow down the right hand branch in the absence of a laser. However, when the water interface crosses the laser spot, it is com-pletely blocked, indicating that there exists a force acting from the right to the left, i.e. coun-teracting the viscous drag produced by the ex-ternal flow. Unlike the usual Marangoni force, this force is directed from the heated region cen-tered on the focusing point of the laser to the cold regions of the flow away from the laser thereby balancing the drag from the external oil flow and blocking the droplet.

Rather than simply ruling out a thermocapil-lary origin of the observed force, we have intro-duced tiny toner particle in the fluids in order to visualize in detail the direction of the flow in our experiments. Figure 2(a) shows that the flow is directed form the cold regions towards the hot spotalong the interface, pointing to an ‘in-verse’ Marangoni effect. Without ignoring that, for pure liquids, the direction of Marangoni flow along the interface is directed from the hot (low surface tension) to the cold (high surface ten-sion) regions, this opposite finding is consistent with previous studies [6, 7] which have shown an increase of surface tension with temperature in the presence of surfactants.

2. GOVERNING EQUATIONS AND NUMERICAL METHOD

Our experiments were conducted in high as-pect ratio channels. The typical dimension in the transverse direction is about 500 µm, while the typical dimension in the thickness is about

laser

(a)

main flow

Figure 2: (a) Overlay of 100 images from a video sequence, showing the motion of seeding parti-cles near the hot spot. Note that the motion along the interface is directed towards the hot spot. Channel width is 140 µm.(b) Streamfunc-tion contours obtained by solving the depth-averaged model described in the text. Red con-tours indicate clockwise flow and blue concon-tours

indicate counter-clockwise flow. h/R = 0.2,

w/R = 0.5.

20-30 µm. With typical velocities of 1mm/s, the Reynolds number is seen to be very small. For these two reasons, we use a Hele-Shaw-like approximation, based on the depth-averaged Stokes equations as Boos and Thess [8], Bush [9] or Nadim et al. [10]. Note that, contrary to the Hele-Shaw approximation, where the depth-averaged velocity field derives from a potential and cannot fulfill the Marangoni induced tan-gential shear stress discontinuity at the inter-face, higher-order terms are kept in the

equa-tions to yield a 4th_{-order biharmonic-type}

er eθ h y x z R

Figure 3: Schematic of the depth-averaged

model.

Let us consider a circular drop of radius R in an infinite domain, as sketched in Fig 3. As-suming a parabolic profile in the small dimen-sion (z) and introducing a streamfunction for the mean velocities in the plane of the channel, the depth averaged equations derived from the Stokes equations valid in each fluid are

 1 r ∂ ∂rr ∂ ∂r+ 1 r2 ∂2 ∂θ2   1 r ∂ ∂rr ∂ ∂r+ 1 r2 ∂2 ∂θ2 − 12 h2  ψ = 0, (1) where the depth-averaged velocities may be

retrieved from uθ = −∂ψ/∂r and ur =

1/r(∂ψ/∂θ). At the drop interface (r = R), two kinematic boundary conditions are imposed; zero normal velocity

ψ(1)_{= ψ}(2)_{= 0, (2)}

and the continuity of the tangential velocity

∂ψ(1)

∂r =

∂ψ(2)

∂r . (3)

The normal dynamic boundary condition is not imposed since the geometry of the drop is imposed and the transverse curvature of the meniscus is assumed to counterbalance the pres-sure difference at the interface. Finally, the tan-gential dynamic boundary condition which takes into account the Marangoni stress is

µ1r ∂ ∂r  u1 θ r  − µ2r ∂ ∂r  u2 θ r  = −γ_r′dT_dθ, (4)

where µ1,2 are the dynamic viscosities and u1,2_θ

are the azimuthal velocities in the drop and

the carrier fluid, respectively, and where γ′ ₌

∂γ/∂T is the surface tension to temperature gra-dient and T (r, θ) is the imposed temperature field.

A pseudo-spectral discretization technique combining a Tchebitscheff collocation method in

the radial direction with respectively Nr(1) and

Nr(2) collocation points and a Fourier expansion

along the azimut with Nθmodes is implemented

in each fluid together with the 4 boundary con-ditions at the interface. The symmetry condi-tions at the origin in the inner fluid are taken implicitly into account as in [11] and an outer

radius of rmax = 8R is chosen to impose the

far field boundary conditions in the outer fluid. The code, written in matlab, can be run on a laptop in a few minutes as long the memory re-quirements remain reasonable (i.e. as long as h/R is not too small, preventing the appear-anceof very thin boundary layers in each fluid). The convergence of the results with increasing resolution has been checked. We use in general

N1

r = 40, Nr2= 40, Nθ= 40.

As a typical test-case, the Marangoni induced flow around a drop in a linear temperature gra-dient and with a positive surface tension to

tem-perature gradient γ′ _{> 0 is considered and our}

results are compared to the analytical solution of [8] in figure 4, displaying an excellent agree-ment, with errors less than 0.1% for h/R = 0.2.

T

∆

Figure 4: Superposition of the analytical solu-tion obtained by Boos and Thess [8] (full iso-lines) and our numerical computation (dashed isolines) in presence of a linear temperature

gra-dient. h/R = 0.2 and µ2 = µ1. The color

con-vention is similar to Fig. 2b.

3. LOCALIZED HEATING

Thess[8] is only valid for a linear temperature gradient and a numerical approach becomes compulsory when the heating is localized, so as to mimic the experiment. According to the ex-perimental observations, the surface tension to

temperature gradient γ′ _{is taken positive and}

may be estimated at γ′ _{∼ 1 mNm}−1_K−1 _[6,

7]. For simplicity, we approximate the steady state temperature distribution using a Gaussian form corresponding to a hot region of size w,

T (x, y) = ∆T exp−(x−R)2 +y2_{w2 , where ∆T is the}

maximum temperature difference between the hot spot and the far field and w corresponds to the hot spot size and is significantly larger

than ω0. The equations are subsequently made

nondimensional using ∆T as temperature scale,

R as length scale, Rγ′_{∆T as force scale and}

µ1+µ2

γ′_∆T as time scale, the remaining

nondimen-sional groups being the aspect ratio h/R, the nondimensional spot size w/R and the viscosity

ratio ¯µ2 = µ2/(µ1+ µ2). Nondimensional

vari-ables are denoted in the sequel by an overbar. A typical flow field is shown in Fig. 2(b). As in the experimental image, and despite the dif-ference in the drop shape and in the presence of lateral walls, four recirculation regions are clearly visible and the flow is seen to converge towards the hot region along the interface.

0 0.2 0.4 0.6 0.8 1 −0.3 −0.2 −0.1 0 w/R pressure stress total force h/R=0.05 h/R=0.1 h/R=0.2 tangential shear normal shear stress F

(c)

Figure 5: Nondimensional force ¯F plotted as a

function of w/R for various aspect ratios h/R

(individual points) for ¯µ2= 3/4. The lines

cor-respond the asymptotic expression Eq. 14. Inset shows the distribution along the azimuthal di-rection of the pressure, normal and tangential shear stresses for h/R = 0.2, w/R = 0.5. Their sum yields the total force.

Three stress terms apply a net force on the

drop: the pressure field as well as the viscous shear stress in the tangential and normal direc-tions. The resulting x-component of the total force on the drop is given by the projections of these three contributions, integrated along the azimuthal direction, as shown in the inset of Fig. 5 for h/R = 0.2 and w/R = 0.5, where these three integrands and their sum are repre-sented (once suitably normalized) as a function of the azimut by the signed distance of the cor-responding curve from the unit circle. Note that the global x-component of the force is negative (the y component vanishes by symmetry) and is thereby oriented away from the laser. It may be shown in addition that the integral of the wall friction vanishes since the drop is stationary.

Let us compute the pressure field as well

as the normal ¯σ¯r¯r = 2¯µ2∂_∂¯u¯_rr¯ and

tangen-tial ¯σrθ¯ = ¯µ2 1_r¯∂_∂θu¯r¯ +∂_∂¯u¯_rθ −u¯_r¯θ

viscous shear

stresses in the external flow on the interface at ¯

r = ¯R = 1.

Since ¯ur¯ vanishes on the interface at ¯r = 1

independently of θ, the viscous tangential shear force reduces to ¯ Ft= −¯µ2 h R Z 2π 0  ∂ ¯uθ ∂¯r − ¯ uθ ¯ R  ¯ r=1 sin(θ) ¯Rdθ = ¯µ2 h R Z 2π 0  −∂ ¯uθ ∂¯r + ¯uθ  ¯ r=1 sin(θ)dθ. (5)

The viscous normal force is reexpressed using

the continuity equation ∂u¯¯r

∂r¯ = − 1 ¯ r ∂¯uθ ∂θ − ¯ u¯r ¯ r and ∂u¯r¯ ∂¯r 

_r=1¯ = −1_r¯∂¯_∂θuθ on the drop interface,

¯ Fn= 2¯µ2 h R Z 2π 0  −1_¯ R ∂ ¯uθ ∂θ  ¯ r=1 cos(θ) ¯Rdθ, (6) and after an integration by part

Fn= −2¯µ2¯h

Z 2π

0

(¯uθ)r=1¯ sin(θ)dθ. (7)

The pressure force is given by ¯ Fp= h R Z 2π 0 − ¯p|r=1¯ cos(θ) ¯Rdθ (8)

An integration by parts first gives ¯ Fp= h R Z 2π 0 ∂ ¯p ∂θ     ¯ r=1 sin(θ) ¯Rdθ (9)

The term ∂ ¯p/∂θ may be retrieved from the

equations ¯ µ2  1 ¯ r ∂ ∂¯r¯r ∂ ∂¯r− 1 ¯ r2+ 1 ¯ r2 ∂2 ∂θ2 − 12R2 h2  ¯ uθ + ¯µ2 2 ¯ r2 ∂ ¯ur¯ ∂θ = 1 ¯ r ∂ ¯p ∂θ. (10)

Since ¯ur¯vanishes on the interface at ¯r = 1

inde-pendently of θ, the term _¯r22

∂u¯(2)r¯

∂θ is zero at ¯r = 1

and the pressure force becomes ¯ Fp = ¯µ2h R Z 2π 0 ¯ R ∂ 2 ∂¯r2+ 1 ¯ R ∂ ∂¯r −_¯1 R2 + 1 ¯ R2 ∂2 ∂θ2 − 12R2 h2  ¯ uθ  sin(θ) ¯Rdθ     ¯ r=1 . (11) A double integration by parts leads

Fp = ¯µ2 h R Z 2π 0  ∂2 ∂¯r2+ ∂ ∂¯r− 2 −12R 2 h2  ¯ uθ  sin(θ)dθ    _¯r=1 . (12) The final expression for the nondimensional force is obtained by adding (5), (7) and (12)

¯ F = ¯Ft+ ¯Fn+ ¯Fp = ¯µ2 h R Z 2π 0  ∂2 ∂¯r2 − 12R2 h2 − 3  ¯ uθsin(θ)dθ    _r=1¯ . (13) Results of the computations for different aspect ratios (h/R) and hot spot size (w/R) and for ¯

µ2 are shown in Fig. 5, where it appears that

the data is well fitted (see the solid lines) by an expression as simple as ¯ F ≃ −α¯µ2 h R w R, (14)

with the prefactor α ≃ 3.5. This expression may be understood by replacing the implicit (since it

involves ¯uθ which is deduced from the solution

of the biharmonic equation (1)-(2)) expression (13) by its explicit boundary layer approxima-tion (in the spirit of [8, 10]) in the limit where h is small compared to all characteristic in-plane length scales (h ≪ R, w) ¯ F ≃ −2¯µ2 h R Z 2π 0 ¯ T (¯r = 1, θ) cos(θ)dθ. (15)

This expression, which requires only the knowl-edge of the temperature profile, may be further approximated in the limit of small w/R by

¯

F ≃ −2√π ¯µ2h

R w

R, (16)

in fine agreement with the fitted value of α since

2√π ≃ 3.55. The resulting dimensional scaling

is easily retrieved

F ≃ −2

√

π∆T γ′_hwµ2

R(µ1+ µ2) . (17)

With µ1 = 10−3 Nm−2s (water) and µ2 = 3µ1

(hexadecane), the calculated force is on the or-der of 0.1 µN, a very large force compared with optical tweezers.

4. CONCLUSION

In summary, we have experimentally and theoretically illustrated the efficiency of laser-driven localized thermocapillary stresses for mi-croactuation of water droplets. By locally heat-ing the oil-water interface, a net “pushheat-ing” force is produced, that may be used in various prac-tical microfluidic applications: Controlling the timing of drop formation, controlling the size of the drop and the path that it follows when ar-riving at a bifurcation. The generality of the process provides a practical new way for acting on individual droplets, at any location, while working inside the robust environment of the microchannel.

Several scaling laws are favorable to this forcing technique: The dominance of surface stresses over volumetric effects at small scales and the rapidity of diffusive heat transfer while thermal inertia is reduced all lead to very rapid response time. Moreover, the force due to the laser heating decreases as 1/R with the drop ra-dius R (Eq. 17) while the drag force [10] scales

as R2_{. Therefore the laser power necessary to}

counterbalance the drag force decreases with the drop size, which also favors miniaturization.

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